First slide

Simple harmonic motion

Question

The potential energy of a particle executing SHM along the x-axis is given by $U = U_{0} - U_{0} \cos a x$ . What is the period of oscillation?

Moderate

A

$2 π \sqrt{\frac{m a}{U_{0}}}$
B

$2 π \sqrt{\frac{U_{0}}{m a}}$
C

$\frac{2 π}{a} \sqrt{\frac{m}{U_{0}}}$
D

$2 π \sqrt{\frac{m}{a U_{0}}}$

Solution

$\begin{array}{l} U = U_{0} - U_{0} \cos a x \\ F = - \frac{d U}{d x} = - U_{0} a \sin a x = m \frac{d^{2} x}{d t^{2}} \end{array} F o r s m a l l v a l u e s o f x, m \frac{d^{2} x}{d t^{2}} = - U_{0} a^{2}$
$This is the equation of SHM \Rightarrow T = \frac{2 π}{a} \sqrt{\frac{m}{U_{0}}}$

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